The number of real roots of the equation 5 + |2x -- 1| = 2x (2x -- 2) is

Question

The number of real roots of the equation
 5 + |2x
-- 1| = 2x
(2x
-- 2) is

Accepted Answer

When 2x $$ \ge $$ 1

5 + 2x --1 = 2x (2x -- 2)

Let 2x = t

$$ \Rightarrow $$5 + t -- 1 = t (t -- 2)

$$ \Rightarrow $$ t = 4, -- 1(rejected)

$$ \Rightarrow $$ 2x = 4

$$ \Rightarrow $$ x = 2

Now when 2x < 1

5 + 1 -- 2x = 2x (2x -- 2)

Let 2x = t

$$ \Rightarrow $$ 5 + 1 -- t = t (t -- 2)

$$ \Rightarrow $$ 0 = t2 -- t -- 6

$$ \Rightarrow $$ 0 = (t -- 3) (t -- 2)

$$ \Rightarrow $$ t = 3, -- 2

2x = 3, 2x = -- 2 (rejected)

$$ \therefore $$ Only one real roots.

Solutions · 1